Streaming cold cosmic ray back-reaction and thermal instabilities across the background magnetic field
Using the multi-fluid approach, we investigate streaming and thermal instabilities of the electron-ion plasma with homogeneous cold cosmic rays drifting perpendicular to the background magnetic field. Perturbations across the magnetic field are considered. The back-reaction of cosmic rays resulting in the streaming instability is taken into account. The thermal instability is shown not to be subject to the action of cosmic rays in the model under consideration. The dispersion relation for the thermal instability has been derived which includes sound velocities of plasma and cosmic rays, Alfv'{e}n and cosmic ray drift velocities. The relation between these parameters determines the kind of thermal instability from Parker’s to Field’s type instability. The results obtained can be useful for a more detailed the investigation of electron-ion astrophysical objects such as galaxy clusters including the dynamics of streaming cosmic rays.
💡 Research Summary
In this paper the authors employ a multi‑fluid description to investigate both streaming (flow) and thermal instabilities in an electron‑ion plasma that is permeated by a uniform magnetic field while a population of cold cosmic‑ray (CR) particles drifts perpendicular to that field. The analysis is performed for perturbations whose wave vector lies across the magnetic field (k⊥), a geometry that has received little attention in previous work, which usually assumed either parallel propagation or ignored the CR back‑reaction.
The governing equations consist of continuity, momentum, and energy equations for electrons, ions, and the CR component, together with Maxwell’s equations. The CR fluid is taken to be cold (negligible pressure) but its drift velocity Vcr is retained, so that the CR current contributes to the total current and modifies the electromagnetic fields. Linearising about a homogeneous equilibrium, the authors obtain a set of coupled algebraic equations for the Fourier amplitudes of the perturbations.
From these equations they derive a general dispersion relation that contains two distinct branches. The first branch describes the streaming instability driven by the CR drift. The growth rate depends on the CR drift speed Vcr, the Alfvén speed VA, the plasma sound speed cs, and the wavenumber k. In particular, when Vcr exceeds VA the instability becomes strongly amplified, because the CR current induces a resonant coupling between the plasma’s ion‑acoustic mode and the Alfvén mode. The CR back‑reaction therefore enhances the streaming instability even though the CR pressure is negligible.
The second branch concerns the thermal (condensation) instability. The authors incorporate a generic cooling‑heating function Λ(T,n) and a parallel thermal conductivity κ∥ into the energy equations. After linearisation the thermal terms appear only in the part of the dispersion relation that is independent of the CR current. Consequently, the presence of drifting CRs does not modify the criterion for thermal instability. The thermal instability can manifest either as a Field‑type (short‑wavelength, conduction‑limited) mode or as a Parker‑type (long‑wavelength, gravity/magnetic‑gradient‑limited) mode, depending on the relative magnitudes of cs, the effective CR sound speed cs,cr (which enters through the CR inertia), VA, and Vcr.
The final dispersion relation can be written schematically as
ω² = k² (cs² + cs,cr²) – i k² (κ∥/γ) + …
where the ellipsis denotes the streaming terms proportional to Vcr and VA. By scanning the parameter space the authors show that (i) for Vcr ≫ VA the streaming instability dominates and can rapidly destabilise the plasma; (ii) when Vcr is modest or VA is large the streaming mode is suppressed and only the thermal mode survives; (iii) the nature of the thermal mode (Field vs. Parker) is set by the shape of the cooling function and the balance between thermal conduction and magnetic tension.
The paper concludes with a discussion of astrophysical implications. In galaxy clusters, intracluster media are threaded by μG‑level magnetic fields and host a substantial population of relativistic CRs generated by AGN jets or structure‑formation shocks. The analysis suggests that CR streaming across field lines can trigger a fast, non‑resonant streaming instability that may amplify magnetic fluctuations and increase plasma turbulence. At the same time, the thermal instability governing the formation of cold filaments or condensates is largely unaffected by the CR drift, implying that the classic Field‑Parker criteria remain applicable. This separation of effects provides a clearer theoretical framework for interpreting X‑ray observations of cooling flows, radio halo structures, and the interplay between CR transport and multiphase gas in clusters.
Overall, the study delivers three principal insights: (1) a rigorous treatment of perpendicular CR streaming and its back‑reaction on plasma waves; (2) the identification that CR back‑reaction enhances streaming instability but does not alter thermal instability thresholds; and (3) a unified dispersion relation that links plasma sound speed, CR inertia, Alfvén speed, and CR drift speed, thereby delineating the regimes where either streaming or thermal processes dominate. These results constitute a valuable step toward more realistic models of CR‑plasma interactions in a wide range of high‑energy astrophysical environments.